Download Simulations of dusty plasmas using a special-purpose

Simulations of dusty plasmas using a special-purpose computer system
for gravitational N-body problems
K. Yamamotoa, Y. Mizunoa, H. Inuzukaa, Y. Caob, Y. Liub, K. Yazawab
a
Faculty of Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu 432-8561, Japan
b
Hamamatsu Metrix Co.,Ltd, 1-4-10-8 shinmiyakoda, 431-2103, Japan
Simulations of dusty plasmas were performed using GRAPE-6, a special-purpose computer
designed for gravitational N-body problems. The collective behaviour of dust particles, which are
injected into the plasma, was studied by means of three-dimensional computer simulations. As an
example of a dusty plasma simulation, we have calculated movement of dust particles in plasmas
under microgravity conditions. In this simulation, a dust-free region called the “void” is observed
in the dust cloud. Another example was to simulate experiments on Coulomb crystals in plasmas.
Formation of a Coulomb crystal was observed under typical laboratory conditions. For the
simulation of a dusty plasma in microgravity with 3x104 particles, GRAPE-6 can perform the
whole operation 1000 times faster than by using a Pentium4-1.6GHz CPU.
1. Introduction
2. Calculations using GRAPE-6
A unit of G6 has 32 custom LSI chips, each
containing 6 pipeline processors for the calculation
of gravitational interactions between particles. Since
one pipeline processor works logically as 8
pipelines, G6 has 1536 pipelines per unit. G6
x, m
Particle
Memory
x, m
Pipeline
Particle
Memory
Fi
Fi
Pipeline
…
…
In order to achieve high calculation speeds, and to
lower the cost of the hardware used for simulations,
special-purpose computers have been developed
recently. A special-purpose computer is typically a
massively parallel processing computer, which has
many custom LSI chips specifically designed to
calculate a few expressions. By limiting the
versatility of the calculation, the hardware
complexity can be reduced drastically. Therefore, a
special-purpose computer can contain a greater
number of processors on one chip compare to a
general-purpose computer. GRAPE-6 (G6) is a
special-purpose computer for gravitational N-body
simulations [1] manufactured by Hamamatsu Metrix
Co., Ltd. The G6 can rapidly calculate the
gravitational force between particles using its
specialized pipeline processors. The peak
performance of a G6 unit reaches 985 Gflops. We
have used G6 for the calculation of Coulomb
interactions, which is the same form of central force
as the gravitational force, to accelerate the plasma
simulation. We have previously developed threedimensional electrostatic simulations using G6, and
performed
simulations
of
plasma
waves,
fluctuations, and diffusions. Currently, we are
investigating developing simulations of dusty
plasma using G6. In addition to the plasma, a dusty
plasma contains micro-particles, i.e. dust particles.
Since micrometer size particles are usually
negatively charged - 103~104e - in a plasma (here e
is the elementary charge), the dusty plasma is easily
“strongly coupled”. If specific conditions are
fulfilled, the dust particles form an ordered
crystalline structure called “Coulomb crystal” or
“plasma crystal” [2,3]. Many interesting features of
dusty plasmas have been elucidated in laboratory
experiments, and various shapes of the Coulomb
crystal have been observed, such as the disk, dome,
and cone [4]. However, in laboratory experiments,
the movement of dust particles and the structure of
the Coulomb crystal are restricted due to gravity. In
order to study dusty plasmas by suppressing the
effect of gravity, experiments under microgravity
conditions were carried out [5]. In this experiment, a
dust-free region called the “void” [6] was observed
in the center of the dust cloud. It was also observed
that a part of the dust particles around the void were
in a crystalline state. In this paper, we present some
simulations of a dusty plasma to simulate these
experiments, to examine the basic concept of
computing using G6, and the actual performance of
G6.
Fig. 1. The architecture of GRAPE-6.
Fi  Gmi  m j
j
rij
(rij2  S 2 ) 3 / 2
rij  x i  x j
(1)
 qi
40
q
j
t = 0[s]
y[cm]
0.5
0
-0.5
-0.5
rij
(rij2  S 2 ) 3 / 2
0.5
0
0.5
t = 1[s]
0.5
0
-0.5
(2)
j
0
x[cm]
-0.5
where xi, xj and mi, mj are the position and the mass
of particle i, j, G is the gravitational constant, and S
is a softening parameter used to suppress any
divergence of the force at rij → 0. On the other hand,
the Coulomb force between charged particles is
expressed by
Fi 
conditions. In this simulation, we have assumed
y[cm]
realizes high-speed calculations using these
pipelines efficiently in parallel. G6 is connected to a
general-purpose host computer (a Pentium 4
personal computer) through a communication
interface. G6 calculates only the force due to
gravitation between particles, and the host computer
performs the other calculations. The user program
runs on the host computer, and it can use G6 only by
calling library functions. The architecture of G6 is
fairly simple from the viewpoint of a user. G6 is
comprised of multiple memory units and pipeline
processors. Each LSI chip has its own memory unit,
and whole particle data is stored distributed to these
memory units. Thus, the calculation of the force,
which is exerted on a particle, is divided among all
LSI chips. Besides, G6 calculates the forces on
different particles in parallel using pipeline
processors on the LSI chip. The architecture of G6
is shown in Fig.1.
The gravitational force calculated by a pipeline of
G6 is given by
(3)
where qi and qj are the respective charges of particle
i and j, and  0 is the electric constant. This equation
shows that we can accelerate computations using G6
for the system including Coulomb interaction, by the
replacement of G, mi and mj in expression (1) with –
1/4  0 , qi and qj. In the case of the simulations
presented the following sections, the host computer
performed integration of the equation of motion,
calculation of charging process and external forces.
G6 performs calculations of interaction between
dust particles. This inter-particle force is calculated
as Coulomb force using expression (3).
3. Simulation results
3.1. Behaviour of dust clouds in microgravity
As an example of dusty plasma simulations using
G6, we have calculated movement of dust particles
which are injected into a plasma under microgravity
Fig.2.
x[cm]
Cross sectional view of temporal
evolutions of the dust cloud at z = 0.
= 0.
capacitively coupled radio-frequency (RF) discharge
as the background plasma, and micron-order radius
spheres as the dust particles. Considering the size of
the dust particles and the plasma conditions, we use
three external forces as the dominant external forces
acting on the dust particles: ion drag force, neutral
gas drag force, and electrostatic confining force [7].
The magnitudes of these external forces and others,
which may play a role under similar conditions,
have been investigated on an experimental basis [8].
The interaction between dust particles is calculated
using the Coulomb interaction shown in expression
(3). The background plasma affects the space charge
quasi-neutral condition, the number of charges on
the dust particle, and magnitude of the external
forces. In this example, we use an argon plasma with
a neutral gas pressure of 0.1 torr and a plasma
density of 1015 ~ 1016 m-3. The electron and ion
temperature was 1 eV and 0.05 eV, respectively. As
an initial condition, dust particles were uniformly
distributed within the system using random
numbers, and the velocity distribution was set to the
Maxwellian velocity distribution with a temperature
of 300 K. Figure 2 shows the temporal evolution of
the spatial distribution of the dust particles. It is
random numbers, and the velocity distribution is set
to the Maxwellian velocity distribution for a
temperature of 300 K. In the simulations, the
Diameter [cm]
seen that dust particles are pushed out of the central
region, and are arranged at the outer edge of the
system. This results in the appearance of a
centimeter-size void inside the dust cloud. It is also
observed that the dust number-density is enhanced
around the void, and the dust particles form a rather
stable structure. This “void formation” has been
observed in some laboratory as well as microgravity
experiments [5,9]. In microgravity, sufficiently large
dust particles can exist within the main plasma,
different from laboratory experiments, and void
formation occurs easily in dusty plasmas. The void
characteristics obtained in the simulations, such as
the void diameter, are similar to those seen in the
microgravity experiments.
In the present simulation, we have also studied
variations in the void diameter caused by varying
external conditions. Figure 3 illustrates the variation
of the void diameter as a function of the plasma
density. It is seen that the void diameter increases
with plasma density. However, in the region where
the plasma density is very low, no void formation is
observed in the simulation. This tendency is
analogous with that seen in some experimental
results [10], where the void size is observed to
increase with input RF power.
6.0
4.0
2.0
0
2.0
4.0
6.0
8.0
15
Plasma density [×10
10.0
-3
m ]
Fig.3. Dependence of the void diameter on plasma
density.
particles move about in the system with thermal
velocity at first, but after a period of time, are
cooled by the neutral gas and form an ordered
crystalline structure. This transition from a fluid
liquid-like phase to a stationary solid-like phase, is
referred to as “Coulomb crystallization”. An
(a)
z[cm]
1.2
1.0
0.8
-0.2 -0.1
0
0.1 0.2
x[cm]
(b)
0.5
y[cm]
3.2. Coulomb crystallization in plasmas
The next example is to simulate experiments of
Coulomb crystallization in plasmas. In this
simulation, we have studied crystallization of the
dust clouds, due to Coulomb interactions between
dust particles. These inter-particle forces are directly
calculated using G6 with expression (3). We have
used a capacitively coupled RF plasma for the
background plasma, and a micron-order-radius
sphere for the dust particles. It is assumed that the
dust particles are levitating in the plasma-sheath
boundary above the electrode. The external
conditions, such as plasma parameters and gas
pressure, are set to typical values of the usual
laboratory experiments. The effects of gravity,
neutral gas drag, and electric field are considered as
external forces exerted on the dust particles. In the
vertical direction, a strong electric field, i.e. the
sheath electric field, is introduced to support the
dust particles against gravity. Besides, there is a
weak confining electric field in a lateral direction to
confine the dust particles. In this example, we used
an argon plasma with a neutral gas pressure of 0.1
torr and a plasma density of 1015 m-3. The electron
and ion temperatures were 3 eV and 0.05 eV,
respectively. As an initial condition, dust particles
were uniformly distributed within the system using
0
-0.5
-0.5
0
0.5
x[cm]
Fig.4. Crystalline structure of the dust
particles. (a) Side view. (b) Top view.
example of the observed crystalline structure is
shown in Fig.4. Figure 4 (a) shows a side view of
the whole structure, and Fig.4 (b) shows a top view.
From Fig.4 (a), it can be seen that the dust cloud is
divided into several layers in the z-axial direction,
and the dust particles are confined within each layer.
Although some disorder is found in the central
region, a nearly isotropic ordered structure is seen in
Fig.4 (b). In some laboratory experiments, the
structure of the Coulomb crystal has been reported
as containing several layers arranged vertically, with
the dust particles within the layers showing a
uniformly-spaced isotropic arrangement in a lateral
direction [3]. Unlike the laboratory experiments, the
lateral arrangement of dust particles within the
layers was not completely isotropic in the
simulation. However, the overall structure is
analogous with that of the Coulomb crystal observed
in the laboratory experiments.
to those shown by some experimental results. We
have also simulated the Coulomb crystallization in
plasmas. The crystalline structure of the dust
particles is observed in the plasma-sheath boundary.
It was seen that the crystalline structure consisted of
several layers arranged vertically, with a nearly
3.3. Evaluation of the calculation speed of G6
Finally, in order to evaluate the actual calculation
speed of G6 for the simulations, we compare the
calculation speed of G6 and a personal computer.
Figure 5 shows the computing time required to
perform the whole operation of a simulation using
G6, and time using only the host computer
(incorporating a Pentium 4-1.6 GHz CPU). The
simulation of a dusty plasma in microgravity, which
is presented above, is used for the comparison. It is
shown that it takes 12.1 hours to perform the
simulation with 3×104 particles using the Pentium
4 personal computer, and 43 seconds using G6.
This advantage becomes obvious in the case of
simulations containing more than 104 particles. It
was also shown that the whole calculation of the
simulation can be performed at an average speed of
800 Gflops by using the special- purpose computer
G6.
Fig.5. Comparison of the calculation time using
GRAPE-6 with the time required by a
Pentium-4 1.6GHz CPU.
4. Conclusions
We have performed simulations of dusty plasmas
using a special-purpose computer system, the
GRAPE-6. The collective behaviour of dusty
plasmas in microgravity was simulated. It was
observed that a centimeter-size void appeared inside
the dust cloud under typical plasma conditions in
some micro- gravity experiments. The dust particles
around the void were strongly coupled and formed a
stable structure. The simulated void diameter
increased with plasma density. However, when the
plasma density was very low, no void formation was
observed. This behaviour of the dust cloud is similar
Calculation time [s]
105
104
Pentium-4,1.6GHz
103
102
101
100
GRAPE-6
103
104
105
Number of particles
isotropic lateral arrangement of the dust particles
within each layer. The crystalline structure thus
obtained is analogous to that seen in some
laboratory experiments. It was confirmed that the
whole calculation of the simulation can be
performed at an average speed of 800 Gflops by
using the special-purpose GRAPE-6 computer.
Acknowledgement
This work is supported in part by a grant from the
Japan Small and Medium Enterprise Corporation.
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